Density Estimation with Contaminated Data: Minimax Rates and Theory of Adaptation

TL;DR

This paper analyzes the fundamental limits of density estimation under contamination, deriving minimax rates and exploring the costs of adapting to contamination levels and smoothness, with implications for robust statistical inference.

Contribution

It provides the first comprehensive minimax rates for contaminated density estimation and develops adaptation strategies, including variations of Lepski's method, under different contamination models.

Findings

Minimax rate characterized by contamination and smoothness parameters

Optimal adaptation incurs small or no costs for certain parameters

Adaptation to both contamination level and smoothness simultaneously is impossible

Abstract

This paper studies density estimation under pointwise loss in the setting of contamination model. The goal is to estimate $f(x_0)$ at some $x_0\in\mathbb{R}$ with i.i.d. observations, $$ X_1,\dots,X_n\sim (1-\epsilon)f+\epsilon g, $$ where $g$ stands for a contamination distribution. In the context of multiple testing, this can be interpreted as estimating the null density at a point. We carefully study the effect of contamination on estimation through the following model indices: contamination proportion $\epsilon$, smoothness of target density $\beta_0$, smoothness of contamination density $\beta_1$, and level of contamination $m$ at the point to be estimated, i.e. $g(x_0)\leq m$. It is shown that the minimax rate with respect to the squared error loss is of order $$ [n^{-\frac{2\beta_0}{2\beta_0+1}}]\vee[\epsilon^2(1\wedge m)^2]\vee[n^{-\frac{2\beta_1}{2\beta_1+1}}\epsilon^{\frac{2}{2\beta_1+1}}], $$ which characterizes the exact influence of contamination on the difficulty of the problem. We then establish the minimal cost of adaptation to contamination proportion, to smoothness and to both of the numbers. It is shown that some small price needs to be paid for adaptation in any of the three cases. Variations of Lepski's method are considered to achieve optimal adaptation. The problem is also studied when there is no smoothness assumption on the contamination distribution. This setting that allows for an arbitrary contamination distribution is recognized as Huber's $\epsilon$-contamination model. The minimax rate is shown to be $$ [n^{-\frac{2\beta_0}{2\beta_0+1}}]\vee [\epsilon^{\frac{2\beta_0}{\beta_0+1}}]. $$ The adaptation theory is also different from the smooth contamination case. While adaptation to either contamination proportion or smoothness only costs a logarithmic factor, adaptation to both numbers is proved to be impossible.